Mathematics for the interested outsider

Modules

With the group algebra in hand, we now define a “-module” to be a module for the group algebra of . That is, it’s a (finite-dimensional) vector space and a bilinear map . This map must satisfy and .

This is really the same thing as a representation, since we may as well pick a basis for and write . Then for any we can write

That is, is a linear map from to itself, with its matrix entries given by . We define this matrix to be , which must be a representation because of the conditions on above.

Conversely, if we have a matrix representation , we can define a module map for as

where we apply the matrix to the column vector . This must satisfy the above conditions, since they reflect the fact that is a representation.

In fact, to define , all we really need to do is to define it for the basis elements . Then linearity will take care of the rest of the group algebra. That is, we can just as well say that a -module is a vector space and a function satisfying the following three conditions:

is linear in : .

preserves the identity: .

preserves the group operation: .

The difference between the representation viewpoint and the -module viewpoint is that representations emphasize the group elements and their actions, while -modules emphasize the representing space . This viewpoint will be extremely helpful when we want to consider a representation as a thing in and of itself. It’s easier to do this when we think of it as a vector space equipped with the extra structure of a -action.

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For whatever reason, I took a course in Module Theory at Caltech, got lost in the forest, and couldn’t remember by the end what I was learning this FOR. Thanks for being so clear. Maybe I’ll get it better this time around…

[…] course, this shouldn’t really surprise us. After all, representations of are equivalent to modules for the group algebra; and the very fact that is an algebra means that it comes with a bilinear […]

[…] way of looking at it: remember that a representation of a group on a space can be regarded as a module for the group algebra . If we then add a commuting representation of a group , we can actually […]

[…] that we’re interested in concrete actions of Lie algebras on vector spaces, like we were for groups. Given a Lie algebra we define an -module to be a vector space equipped with a bilinear function […]

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This is mainly an expository blath, with occasional high-level excursions, humorous observations, rants, and musings. The main-line exposition should be accessible to the “Generally Interested Lay Audience”, as long as you trace the links back towards the basics. Check the sidebar for specific topics (under “Categories”).

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